A005430 Apéry numbers: n*C(2*n,n). (Formerly M2028)

0, 2, 12, 60, 280, 1260, 5544, 24024, 102960, 437580, 1847560, 7759752, 32449872, 135207800, 561632400, 2326762800, 9617286240, 39671305740, 163352435400, 671560012200, 2756930576400, 11303415363240, 46290177201840, 189368906734800, 773942488394400

Offset

0,2

Comments

Sum_{n >= 1} 1/a(n) = Pi*sqrt(3)/9 - Benoit Cloitre, Apr 07 2002

Appears as diagonal in A003506. - Zerinvary Lajos, Apr 12 2006

The aerated sequence 1,0,2,0,12,0,60,0,... has e.g.f. 1+x*Bessel_I(1,2x). - Paul Barry, Mar 29 2010

Conjecture: the terms of the inverse binomial transform are 2*A132894(n). - R. J. Mathar, Oct 21 2012

References

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.25).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..200

L. Alonso and E. M. Reingold, Analysis of Boyer and Moore's MJRTY Algorithm, 2012.

T. Amdeberhan and Henri Cohen, Bernoulli sum meets golden number, MathOverflow, version of 2017-06-15.

Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.

Benjamin Ruoyu Kan, Polynomial Approximations for Quantum Hamiltonian Complexity, Bachelor's thesis, Harvard Univ., 2023.

H. J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006. Published version on Walking into an absolute sum, The Fibonacci Quarterly, 40(2):175-180, May 2002.

A. J. van der Poorten, A proof that Euler missed...Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/1979), 195-203.

I. J. Zucker, On the series Sum(k>=1) C(2k,k)^(-1)*k^(-n) and related sums, J. Number Theory 20 (1985), no. 1, 92-102.

Wadim Zudilin, An elementary proof of Apery's theorem, arXiv:math/0202159 [math.NT], 2002.

Formula

a(n) = A002011(n-1)/2 = 2 * A002457(n-1).

G.f.: 2*x/sqrt((1-4*x)^3). - Marco A. Cisneros Guevara, Jul 25 2011

E.g.f.: a(n) = n!* [x^n] exp(2*x)*2*x*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Peter Luschny, Aug 25 2012

D-finite with recurrence (-n+1)*a(n) + 2*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012

G.f.: 2*x*(1-4*x)^(-3/2) = -G(0)/2 where G(k) = 1 - (2*k+1)/(1 - 2*x/(2*x - (k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012

a(n-1) = Sum_{k=0..floor(n/2)} k*C(n,k)*C(n-k,k)*2^(n-2*k). - Robert FERREOL, Aug 29 2015

From Ilya Gutkovskiy, Jan 17 2017: (Start)

a(n) ~ 4^n*sqrt(n)/sqrt(Pi).

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(phi)/sqrt(5) = A086466, where phi is the golden ratio. (End)

1/a(n) = (-1)^n*Sum_{j=0..n-1} binomial(n-1,j)*Bernoulli(j+n)/(j+n) for n >= 1. See the Amdeberhan & Cohen link. - Peter Luschny, Jun 20 2017

1/a(n) = Sum_{k=0..n} (-1)^(k+1)*binomial(n,k)*HarmonicNumber(n+k) for n >= 1. - Peter Luschny, Aug 15 2017

Maple

A005430 := n -> n*binomial(2*n, n);

Mathematica

Table[n*Binomial[2n, n], {n, 0, 30}] (* Harvey P. Dale, May 29 2015 *)

Prog

(PARI) a(n)=-(-1)^n*real(polcoeff(serlaplace(x^2*besselh1(1, 2*x)), 2*n)) \\ Ralf Stephan
(Magma) [n*Binomial(2*n, n): n in [0..30]]; // G. C. Greubel, Dec 09 2018
(Sage) [n*binomial(2*n, n) for n in range(30)] # G. C. Greubel, Dec 09 2018
(GAP) List([0..30], n-> n*Binomial(2*n, n)); # G. C. Greubel, Dec 09 2018

Crossrefs

Cf. A002736, A005258, A005259, A005429. 1/beta(n, n+1) in A061928.

Cf. A001803, A003506.

Sequence in context: A009618 A143770 A062478 * A094434 A001574 A074445

Adjacent sequences: A005427 A005428 A005429 * A005431 A005432 A005433

Keyword

nonn,easy,nice

Author

Simon Plouffe

Extensions

More terms from James A. Sellers, May 01 2000

Status

approved